Abstract
Discrete Fourier Transform may well be the most promising track in recent music theory. Though it dates back to David Lewin’s first paper (Lewin, J. Music Theory (3), 1959) [33], it was but recently revived by Quinn in his PhD dissertation in 2005 (Quinn, Perspectives of New Music 44(2)–45(1), 2006–2007) [35], with a previous mention in (Vuza, Persp. of New Music, nos. 29(2) pp. 22–49; 30(1), pp. 184–207; 30(2), pp. 102–125; 31(1), pp. 270–305, 1991–1992) [40], and numerous further developments by (Andreatta, Agon, (guest eds), JMM 2009, vol. 3(2). Taylor and Francis, Milton Park) [5], (Amiot, Music Theory Online, 2, 2009) [8], (Amiot, Rahn, (eds.), Perspectives of New Music, special issue 49 (2) on Tiling Rhythmic Canons) [9], (Amiot, Proceedings of SMCM, Montreal. Springer, Berlin, 2013) [10], (Amiot, Sethares, JMM 5, vol. 3. Taylor and Francis, Milton Park (2011) [16], (Callender, J. Music Theory 51(2), 2007) [17], (Hoffman, JMT 52(2), 2008) [29] (Tymoczko, JMT 52(2), 251–272, 2008) [38], (Tymoczko, Proceedings of SMCM, Yale, pp. 258–272. Springer, Berlin, 2009) [39], (Yust, J. Music Theory 59(1) (2015) [42]. I chose to broach this subject because I have had a finger in most, or all, of the pies involved (even using Discrete Fourier Transform without consciously knowing it, in the study of rhythmic tilings).
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Notes
- 1.
I use the modern terms.
- 2.
In the case of mosaic tilings by translation.
- 3.
At the time, probably the only theorist to mention Lewin’s use of DFT.
- 4.
Except for \({\mathcal {F}}_{A}(0)\), which is equal to the cardinality of A.
- 5.
The general definition of \(f*g\) is the map \(t\mapsto \sum \limits _{k\in \mathbb {Z}_n} f(k) g(t-k)\).
- 6.
For instance \(j=(0,1,0,\dots 0)\) is the spectral unit that turns any pc-set A into its translate \(A+1\). Its Fourier coefficients are all \(n^{th}\) roots of unity.
- 7.
There are 6,192 such spectral units for \(n=12\).
- 8.
With the added technical condition \({\mathcal {F}}_{A}(0){\mathcal {F}}_{B}(0)=\#A \#B = n\).
- 9.
In layman’s terms, this means that if motif A tiles, then so does \(\alpha \times A \mod n\), for any \(\alpha \) coprime with n. This is actually a deep algebraic property, but nonetheless it was rediscovered independently by several music composers.
- 10.
At the time the authors made use of polynomials, not Fourier coefficients, but this is an isomorphic point of view. We translated their definitions accordingly.
- 11.
- 12.
There is a good correlation between this saliency and the saturation of the collection in interval d (Aline Honing, personal communication).
- 13.
For 3-sets, \(|{\mathcal {F}}_{A}(3) | = 3 - \dfrac{\pi ^2}{8} V L^2 + o(V L^4)\), best near the maximum, whereas the linear regression yields \( |{\mathcal {F}}_{A}(3)| \approx 3.39 - 1.57 \times V L\). The formula is different from the one in [39] because of a different convention in the definition of DFT.
- 14.
The \(3^rd\) and \(5^{th}\) were chosen for stringent reasons. It was also the choice independently made by [42]).
- 15.
Please remember that this picture is a torus, i.e. opposite sides should be construed as glued together.
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My heartiest thanks to the organizers of this beautiful event for the opportunity of exposing this rich subject to a learned audience.
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Amiot, E. (2017). A Survey of Applications of the Discrete Fourier Transform in Music Theory. In: Pareyon, G., Pina-Romero, S., Agustín-Aquino, O., Lluis-Puebla, E. (eds) The Musical-Mathematical Mind. Computational Music Science. Springer, Cham. https://doi.org/10.1007/978-3-319-47337-6_3
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